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3 (% style="color: rgb(0,0,128);" title="" %)**Mathematics - Numerical Analysis and Optimization**
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5 (% style="color: rgb(0,0,128);" title="" %)**5JUCSN01**
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7 **ECTS**
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9 **3**
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11 **SEMESTER**
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13 **5**
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16 lectures
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18 classes / seminars
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20 practical work
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22 integrative teaching
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24 independent work
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27 10h
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35 20 h
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38 **Language used
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41 French
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44 \\\\
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47 **Course supervisor(s)
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51 (((
52 [[Antoine Henrot>>url:https://iecl.univ-lorraine.fr/membre-iecl/henrot-antoine/||rel="nofollow" shape="rect" class="external-link"]], Professor, [[antoine.henrot@mines-nancy.univ-lorraine.fr>>mailto:antoine.henrot@mines-nancy.univ-lorraine.fr||rel="nofollow" shape="rect" class="external-link"]]
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56 **Key words
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59 (% style="color: rgb(0,0,0);" %)Numerical analysis, optimization, simulation, linear algebra, approximation,** **MATLAB
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62 **Prerequisites**
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64 Basic Mathematics (calculus, linear algebra)
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67 (% title="" %)**Overall objective
68 **
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71 This course is designed to provide a basic understanding of the study and analysis of numerical methods useful for engineering sciences. Various topics will be discussed in lectures and put into practice during tutorials in the Matlab environment. Applications on concrete problems will be developed in order to highlight the importance of mastering numerical simulation, from theory to practice, for the engineer.
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74 (% title="" %)**Course content and organisation
75 **
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78 Session 1: Solving nonlinear equations
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80 Classical algorithms (dichotomy, secant, Newton, fixed point) in dimension 1 and their adaptation in higher dimension. Study of convergence and comparison of algorithms. Methods for accelerating convergence (Aitken, Steffensen).
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82 Session 2: Interpolation and approximation of functions
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84 Lagrange interpolation, Newton formulas (divided differences). Error formulas. Uniform and least squares approximation, cubic spline functions.
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86 Session 3: Integration and numerical derivation
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88 Composite methods of the Newton-Cotes type and Gauss methods. Error analysis. Numerical derivation and error formula. Application to the solution of partial differential equations by finite difference methods.
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90 Sessions 4 and 5: Differential equations
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92 Some theoretical reminders. Numerical solution of differential equations: one-step, explicit and implicit methods (Euler, Crank-Nicholson, Runge-Kutta). Multi-step methods (Adams and its variants). Step adjustment. Notion of stability and convergence. Shooting method for boundary problems.
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94 Sessions 6 and 7: Solving linear systems
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96 Direct methods: Gauss pivot, LU and Choleski factorization. QR factorization. Conditioning of a matrix and effect on the error. Iterative methods (Jacobi, Gauss-Seidel, relaxation, conjugate gradient with or without preconditioning). Convergence and comparison of the different methods.
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98 Session 8 and 9: Optimization
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100 Theoretical basis of optimization: existence of solutions, convexity, first and second order optimality conditions, Lagrange and Karush-Kuhn-Tucker multipliers. Unconstrained optimization, gradient and conjugate gradient methods. Constrained optimization: projection and penalization methods.
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103 (% title="" %)
104 **Skills
105 **
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108 (% style="color: rgb(0,112,192);" %)**Levels**
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110 (% style="color: rgb(0,112,192);" %)**Description and operational verbs**
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113 **Know**
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115 The fundamental aspects of numerical analysis and optimization for engineering and scientific calculus
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118 **Understand**
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120 Numerical simulations as a prediction tool, with its qualities as well as its limits
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123 **Apply(% style="color: rgb(0,112,192);" %) (%%)**
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125 Resolving the practical issues related to engineering development with algorithmic methods
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128 **Analyse(% style="color: rgb(0,112,192);" %) (%%)**
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130 Detect and deduce the properties of certain numerical phenomena using mathematical reasoning.
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133 **Summarise**
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135 Formulate and develop an answer to the problems posed, organize the results into a coherent, rigorous and clear whole.
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138 **Assess**
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140 Judge the relevance of a result and its veracity. Validate the correctness of a method and a reasoning.
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143 (% class="sc-14kwckt-9 kYiyGk" %)**Compliance with the United Nations Sustainable Development Goals
144 **
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